Macquarie Generation CO2 Case Thrown Out

Carbon Central reports that NSW Land and Environment Court has thrown out most of Australia’s first climate change case, agreeing with Macquarie Generation its licence allows it to emit CO2.

NSW Environment Defender’s Office (EDO) argued that the Bayswater coal-fired power station breached its operating licence under the NSW Protection of the Environment Operations Act (POEO) by negligently emitting CO2.

Justice Pain ruled the POEO Act gave it “implied authority” to emit CO2. The licence to burn coal to generate electricity “would have no sensible operation if the licence is construed as not allowing the emission of CO2″, Justice Pain said.

Still, Justice Pain agreed the court should hear EDO’s argument, “novel” to the court’s jurisdiction, that Bayswater was not authorised to emit unlimited amounts of CO2 into the atmosphere, based on the common law “principle of limitation of statutory authority”, used in negligence and/or tort actions involving pollution.

So Macquarie are allowed to emit more than zero but less than an unlimited amount — er, no problem there. They would have to be happy with the result of a frivolous litigation, whose admitted purpose was to merely to allow EDO “to ventilate our argument about CO2 by a different means”.

Continue reading Macquarie Generation CO2 Case Thrown Out

Problem 2 of Climate Science

Problem 2. Cointegration was developed in economics to deal with a problem of spurious correlation between series with stochastic trends. Why should spurious correlation be a concern if the trends in temperature and GHGs are deterministic?

Sometimes I’ve been accused of over-simplifying, but I do try to make models as simple as possible, because it avoids a lot of speculation. With that view, this simple model represents paradoxical features of unit roots. Even if there was a deterministic relation of temperature and CO2, the correlation is spurious.

This simple model shows why temperature can look like a random walk but not go off to infinity.

The model simply includes a reaction to disequilibrium, like a half-pipe – that’s it.

There are a few ways you could get to this model. You could assume a restoring ‘force’ — a second derivative — is a linear function of the anomaly temperature T (difference from equilibrium). Or you could invoke Stefan-Boltzmans Law which states that radiant flux increases with the fourth power of temperature.

The model consists of the non-linear restoring effect f=b*T which is some fraction of T with remembered ‘shocks’ e. The simplification follows:

T(t+1) = T(t)+e-b*T(t)
T(t+1) = (1-b)*T(t)+e

Without f (or b=0), T would this would be a simple random walk RW (black). With f (and b=0.01 my model) (shown in green) the series T (red) has a slight restoring effect.

fig1

RW Dickey-Fuller = -3.0692, Lag order = 4, p-value = 0.1337
T – Dickey-Fuller = -2.9767, Lag order = 4, p-value = 0.1721

It doesn’t take many runs to find one that looks like current global temperature, and the random walk vs T is almost identical over 100 points. Moreover, the adf.test does not reject the presence of a unit root in T (is not stationary).

However, their real difference becomes clear with more data. At 1000 points the random walk (RW) is unbridled, while the T drifts off, it eventually crosses zero again (is mean-reverting).

fig2

Dickey-Fuller = -2.4645, Lag order = 9, p-value = 0.3817
Dickey-Fuller = -3.7678, Lag order = 9, p-value = 0.02060

The adf test now correctly shows that T is stationary, shown by the low p-value, while the RW is not rejected. The series contains a ‘fractional unit root’ of 0.99 instead of one. This difficult to detect difference is enough to guarantee the series reverts to the mean.

The probability distributional is also instructive. The histogram of temperatures has broad flat distribution (red), as would be expected from ‘surfing the half-pipe’, created by the restoring effect (shown in green).

fig3

Moreover, if you squint you can see periodicity in the series. This is because the series hangs around the upper or lower limit for a while, before eventually moving back. The period is only an illusion though. Could this the the origin of PDO?

Also, it’s not hard to see a ‘break’ in the series in the first figure — another illusion.

The distribution of these temperatures is in a broad and flat ‘niche’. Within that ‘niche’ the series is relatively unconstrained, like a random walk, but responding deterministically to small forcings (like CO2?). It is only with more data that the half-pipe is apparent.

It seems to me, that the problem for the deterministic paradigm is that even if CO2 increases temperature deterministically, this deterministic relationship breaks down as the temperature hits the sides and encounters resistance from the non-linear response.

At most, as CO2 keeps increasing, temperature would stay pinned to the side of the bowl, surfing up and down at the limit. When locked into the deterministic view, you would be wondering ‘where is the heat going‘ as extrapolation from increasing CO2 fails.

Global temperature can look like a unit root for all purposes over the last 150 years, but even a small negative feedback in a random walk provides physically-realistic behaviour.

R Code

sim1<-function() {
l<-1000
plot(rep(0,l),ylab="Anomaly",type="l",ylim=c(-10,10))
a=0;b=0.01
v=NULL;d=NULL
r<-rnorm(l,sd=0.2)
c<-cumsum(r)
for (i in r) {
x<-b*a^3
a=a+i-x
v=c(v,x)
d<-c(d,a)
}
lines(c)
lines(d,col=2)
lines(v,col=3)
print(adf.test(c))
print(adf.test(d))
}

Best-Fit Integrated Model of Global Temperature

The acronym ARIMA stands for “Auto-Regressive Integrated Moving Average.” Random-walk and random-trend models, autoregressive models, and exponential smoothing models (i.e., exponential weighted moving averages) are all special cases of ARIMA models. An ARIMA model is classified as an “ARIMA(p,d,q)” model, where the current value y is determined by:

* p — the number of lagged terms (AR),
* d — the number of integrations, and
* q — the number of moving average terms (MA).

Here is a good introduction to ARIMA modelling. Normally, models are either only-AR or only-MA terms, because including both kinds of terms in the same model can lead to overfitting of the data and non-uniqueness of the coefficients.

Paul_K writes

VS proposed an ARIMA (3,1,0) or AR model. B&V concluded an ARIMA((0,1,2) or MA model. I downloaded the GISSTEMP dataset (J-D average), and force fitted both models using a brute-force non-linear solver to minimize RSS in both instances. (I also tested and confirmed VS’s rejection of a drift term in the AR model, not only for his three term solution but also for all combinations of a two-lag solution.)

The GISSTEMP dataset, 1880 to 1909 consists of 130 datapoints with a variance of 0.062388. The best-fit ARIMA(3,1,0) model looks like this(the lower plot being the residual error):

image002

For comparison, the B&V model looks like this:

Continue reading Best-Fit Integrated Model of Global Temperature

Problem 5 of Climate Science

Problem 5. Why do most of the forecasts of climate science fail?

If climate science had a history of accurate forecasts, it would have a foundation for greater credibility. That is what is expected in other fields. Instead, it is “denialist” to say that climate science has a lousy record of predictions.

When I started analysing ecological models in my doctoral studies, it wasn’t ideologically unsound to say that the models did a lousy job, and I spent 3 years trying to work out why. Wouldn’t you think that something could be learned by diagnosing why predictions fail, and coming up with solutions?

What do these examples have in common?

Example 1. Arctic Ice
June 5th, 2009: On Climate Progress, NSIDC director Serreze explains the “death spiral” of Arctic ice, (and the “breathtaking ignorance” for blogs like WattsUpWithThat).
April 4, 2010: Dr. Serreze said this week in an interview with The Sunday Times: “In retrospect, the reactions to the 2007 melt were overstated.” (not breathtakingly ignorant?)

Example 2. Australian Drought
Sept 8th 2003: Dr James Risbey, Monash University: “That means in southern Australia we’d see more or less permanent drought conditions…”
Jan 2009-10: Less permanent drought conditions …

Continue reading Problem 5 of Climate Science

Problem 3 of Climate Science

Problem 3. Why is the concept of ‘climate’ distinguished from the concept of ‘weather’ by an arbitrary free parameter, usually involved in averaging or smoothing or ’scale’ transformations of 10 to 30 years?

The recent article on Question #9 by Meiers and response by Stephen Goddard used a coin toss analogy to answer this question. Meiers states that while the uncertainty of the probability of heads in the short term is high, over the long term we expect the answer to become more certain as we average over more examples. This, he says, is an analogy to weather and climate, and why climate models can be poor at predicting the immediate future, yet can be reliable in the more distant future.

A series containing a unit root contradicts this intuition.

To illustrate I have developed an excel spreadsheet DStrends with a deterministic and stochastic trend (call them I(0) deterministic and I(1) stochastic). It generates a new set of random numbers and trends every time it is reset.

Given a coin toss at time t of Y(t) with equal probability (values -1, 0 and 1) the two series are given by:

D(t) = 0.01t + Y(t)
S(t) = 0.01 + Y(t) + S(t-1)

There is an underlying trend of 0.01t in each series. The two series are quite different as can be seen in the figure below. While the deterministic series shows a slight upward trend, the trend is drowned out by the variability in the series with the unit root.

DStrends

The uncertainty in each series can be seen by plotting the standard error of the mean of each series for increasing observations. The standard error is given by the standard deviation divided by the square root of the number of observations, and represents the uncertainty in the estimation of the mean value.

stderr

The standard error of the deterministic series starts high and decreases. With a low trend (0.01 per year) the uncertainty will keep decreasing. With higher trends (0.1 per year) there will be a minimum around thirty observations (years?) before increasing again as the trend becomes more important.

The standard error of the stochastic series is constant over the range of observations (run the spreadsheet a number of times for yourself). Constant uncertainty is theoretically predicted for a series with a unit root, no-matter what scale we look at the observations.

This illustrates clearly that the notion of uncertain ‘weather’ and more certain ‘climate’ only makes sense for a deterministic series (as shown by the local minimum of uncertainty around 30 observations). There is no sense in a weather/climate distinction when a unit root is present as there is no scale over which estimates of the mean become less uncertain.

So why is there a distinction between weather and climate? The answer is because of the assumption that global temperature is a deterministic trend. However, all the empirical evidence points to the overwhelming effects of a unit (or near unit) root in the global temperature series, which means the deterministic trend is a false assumption.

Dr Meiers proposed a bet which illustrates his erroneous assumption:

How can a more complex situation be modeled more easily and accurately than a simpler situation? Let me answer that with a couple more questions:1. You are given the opportunity to bet on a coin flip. Heads you win a million dollars. Tails you die. You are assured that it is a completely fair and unbiased coin. Would you take the bet? I certainly wouldn’t, as much as it’d be nice to have a million dollars.2. You are given the opportunity to bet on 10000 coin flips. If heads comes up between 4000 and 6000 times, you win a million dollars. If heads comes up less than 4000 or more than 6000 times, you die. Again, you are assured that the coin is completely fair and unbiased. Would you take this bet? I think I would.

If the question was, “Is the coin biased?” and the observable (like temperature) was the final value of an integrating variable I(1), then Dr Meiers’ bet after 10000 would actually be very unwise, as the value of the observable (positive or negative) after 10000 flips is no more certain than after 1 or 100.

Continue reading Problem 3 of Climate Science

Problem 1 of Climate Science

Problem 1. If temperature is adequately represented by a deterministic trend due to increasing GHGs, why be concerned with the presence of a unit root?

Rather than bloviate over the implications of a unit root (integrative behavior) in the global temperature series, a more productive approach is to formulate an hypothesis, and test it.

A deterministic model of global temperature (y) and anthropogenic forcing (g) with random errors e is:

yt=a+b.gt

An autoregressive model of changes in temperature Δyt uses a difference equation with a deterministic trend b.gt-1 and the previous value of y or yt-1:

Δyt =b.gt-1+c.yt-1

Written this way, the presence of the unit root in an AR1 series y is equivalent to the coefficient c equaling zero (see http://en.wikipedia.org/wiki/Dickey%E2%80%93Fuller_test).

I suspect the controversy can be reduced to two simple hypotheses:

H0: The size of the coefficient b is not significantly different from zero.
Ha: The size of the coefficient b is significantly different from zero.

The size of the coefficient should be indicative of the contribution of the deterministic trend (in this case anthropogenic warming) to the global temperature.

We transform the global temperature by differencing (an autoregressive or AR coordinate system), and then fit a model just as we would with any model.

In the deterministic coordinate system, b is highly significant with a strong contribution from AGW. For the AGW forcing I use the sum of the anthropogenic forcings in the RadF.txt file W-M_GHGs, O3, StratH2O, LandUse, and AIE.

fig1


Call: lm(formula = y ~ g)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.34054 0.01521 -22.39 <2e-16 ***
g 0.31573 0.01802 17.52 <2e-16 ***

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1251 on 121 degrees of freedom
Multiple R-squared: 0.7172, Adjusted R-squared: 0.7149
F-statistic: 306.9 on 1 and 121 DF, p-value: < 2.2e-16

The result is very different in the AR coordinate system. The coefficient of y is not significantly greater than zero (at 95%) and neither is b.

fig2


Call: lm(formula = d ~ y + g + 0)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
y -0.06261 0.03234 -1.936 0.0552 .
g 0.01439 0.01088 1.322 0.1887

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.101 on 121 degrees of freedom
Multiple R-squared: 0.0389, Adjusted R-squared: 0.02302
F-statistic: 2.449 on 2 and 121 DF, p-value: 0.09066

Perhaps the main contribution of AGW is since 1960, so we restrict the data to this period and examine the effect. The deterministic trend in AGW is greater, but still not significant.

fig3


Prob1(window(CRU,start=1960),GHG)
Call: lm(formula = d ~ y + g + 0)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
y -0.24378 0.10652 -2.289 0.0273 *
g 0.03050 0.01512 2.017 0.0503 .

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1149 on 41 degrees of freedom
Multiple R-squared: 0.1284, Adjusted R-squared: 0.08591
F-statistic: 3.021 on 2 and 41 DF, p-value: 0.05974

But what happens when we use another data set. Below is the result using GISS. The coefficients are significant but the effect is still small.

fig4


> Prob1(GISS,GHG)
Call: lm(formula = d ~ y + g + 0)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
y -0.27142 0.06334 -4.285 3.69e-05 ***
g 0.06403 0.01895 3.379 0.00098 ***

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1405 on 121 degrees of freedom
Multiple R-squared: 0.1375, Adjusted R-squared: 0.1232
F-statistic: 9.645 on 2 and 121 DF, p-value: 0.0001298

So why be concerned with the presence of a unit root? It has been argued that while the presence of a unit indicates that using OLS regression is wrong, this does not contradict AGW because the effect of greenhouse gas forcings can still be incorporated as deterministic trends.

I am not 100% sure of this, as the differencing removes most of the deterministic trend that could be potentially explained by g.

If the above is true, there is a problem. When the analysis respects the unit root on real data, the deterministic trend due to increasing GHGs is so small that the null hypothesis is not rejected, i.e. the large contribution of anthropogenic global warming suggested by a simple OLS regression model is a spurious result.

Here is my code. Orient is a functions that matches two time series to the same start and end date.


Prob1<-function(y,g) {
v<-orient(list(y,g))
d<-diff(v[,1]);y<-v[1:(dim(v)[1]-1),1];
g<-v[1:(dim(v)[1]-1),2]
l<-lm(d~y+g+0)
print(summary(l))
plot(y,type="l")
lines((g*l$coef[2]+y[1]),col="blue")
}

Central Problems of Climate Science

Prompted by the interest VS has rekindled in fundamental analysis of the temperature series at Bart’s and Lucia’s blogs, below are a small set of core ‘problems’ facing statistical climate science (CS) — kind of a challenge.

Remember a deterministic trend is one brought about by a changing value of the mean, due to a change in an equilibrium value for example (ie non-stationary). A stochastic trend is due the accumulation of random variations; all parameters are stationary.

Problem 1. If temperature is adequately represented by a deterministic trend due to increasing GHGs, why be concerned with the presence of a unit root?

Problem 2. Cointegration was developed in economics to deal with a problem of spurious correlation between series with stochastic trends. Why should spurious correlation be a concern if the trends in temperature and GHGs are deterministic?

Problem 3. Why is the concept of ‘climate’ distinguished from the concept of ‘weather’ by an arbitrary free parameter, usually involved in averaging or smoothing or ‘scale’ transformations of 10 to 30 years?

Problem 4. Why has a community of thousands or tens of thousands of climate scientists not managed to improve certainty in core areas in any significant way in more than a decade (eg the climate sensitivity caused by CO2 doubling as evidenced by little change in the IPCC bounds)?

Problem 5. Why do so many of the forecasts of CS fail (see C3 for list)?

Continue reading Central Problems of Climate Science